When \(\operatorname{Ric}\ge n-1\) , inequality (REF ) was shown by Lévy and Gromov [1]},
see also Theorem 2.1 of [2]}. When \(M\)  is noncompact and \(\operatorname{Ric}\ge 0\) , inequality (REF ) was shown in dimension three by Agostiniani, Fogagnolo and Mazzieri [3]} and in all dimensions by Brendle [4]}. Meanwhile, Brendle proved in [4]} that (REF ) also holds true when \(\Omega \)  is a compact minimal submanifold of \(M\)  of dimension \(n+2\)  with nonnegative sectional curvature.
